probability distribution functionDistribution2002-04-29 23:15:182007-09-22 10:54:49Definitioncumulative distribution functiondistributionrandom variable\usepackage{amssymb}
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\newcommand{\sequence}[1]{\{#1\}}\section{Definition}
Let $(\Omega, \borel, \mu)$ be a measure space. A {\em probability distribution function} on $\Omega$ is a function $f: \Omega \longrightarrow \reals$ such that:
\begin{enumerate}
\item $f$ is $\mu$-measurable
\item $f$ is nonnegative $\mu$-almost everywhere.
\item $f$ satisfies the equation
$$
\int_{\Omega} f(x)\ d\mu = 1
$$
\end{enumerate}
The main feature of a probability distribution function is that it induces a probability measure $P$ on the measure space $(\Omega, \borel)$, given by
$$
P(X) := \int_X f(x)\ d\mu = \int_{\Omega} 1_X f(x)\ d\mu,
$$
for all $X \in \borel$. The measure $P$ is called the {\em associated probability measure} of $f$. Note that $P$ and $\mu$ are different measures, \PMlinkescapetext{even} though they both share the same underlying measurable space $(\Omega, \borel)$.
\section{Examples}
\subsection{Discrete case} Let $I$ be a countable set, and impose the counting measure on $I$ ($\mu(A) := |A|$, the cardinality of $A$, for any subset $A \subset I$). A probability distribution function on $I$ is then a nonnegative function $f: I \longrightarrow \reals$ satisfying the equation
$$
\sum_{i \in I} f(i) = 1.
$$
One example is the Poisson distribution $P_r$ on $\naturals$ (for any real number $r$), which is given by
$$
P_r(i) := e^{-r} \frac{r^i}{i!}
$$
for any $i \in \naturals$.
Given any probability space $(\Omega, \borel, \mu)$ and any random variable $X: \Omega \longrightarrow I$, we can form a distribution function on $I$ by taking $f(i) := \mu(\{X = i\})$. The resulting function is called the distribution of $X$ on $I$.
\subsection{Continuous case}
Suppose $(\Omega, \borel, \mu)$ equals $(\reals, \borel_\lambda, \lambda)$, the real numbers equipped with Lebesgue measure. Then a probability distribution function $f: \reals \longrightarrow \reals$ is simply a measurable, nonnegative almost everywhere function such that
$$
\int_{-\infty}^\infty f(x)\ dx = 1.
$$
The associated measure has \PMlinkname{Radon--Nikodym derivative}{RadonNikodymTheorem} with respect to $\lambda$ equal to $f$:
$$
\frac{dP}{d\lambda} = f.
$$
One defines the {\em cumulative distribution function} $F$ of $f$ by the formula
$$
F(x) := P(\{X \leq x\}) = \int_{-\infty}^x f(t)\ dt,
$$
for all $x \in \reals$. A well known example of a probability distribution function on $\reals$ is the Gaussian distribution, or normal distribution
$$
f(x) := \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-m)^2/2\sigma^2}.
$$